Rate of convergence towards Hartree dynamics

Abstract

The theoretical foundation of Bose-Einstein condensate (BEC) was established about a century ago, and the first experimental observation of BEC was performed two decades ago. It has been interested in the dynamics of low temperature bosonic gas for interacting-systems. Assuming the initial condition to describe a BEC, i.e., a tensor product of factorized states, it has been asked that the system is in the condensation at time $t\ge0$. It has been known that the Hartree dynamics is a proper candidate for the many-body Schrödinger evolution in the mean-field regime. Many results have been obtained and a special focus is placed on the estimate of the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding Hartree dynamics. One fundamental result asserts that the optimal rate of convergence is of order $1/N$ for any fixed time $t\geq0$ with the two-body interaction potential $V\leq D(1-\Delta)$, which covers the Coulomb potential. In this thesis, we extend this result in three different directions

the first is to relax the condition on interaction potentials by permitting more singular potential. The important lemma is proven by using Strichartz estimate. Moreover, we investigate the time dependence of the difference. To have sub-exponential bound, we use the results of time-decay estimate for small initial data. We also refine time-dependent bound for singular potential. For the time dependence we consider interaction potential $V(x)$ of type $\lambda\exp(-\mu|x|)|x|^{-\gamma}$ for $\lambda\in\mathbb{R}$, $\mu\geq0,$and $0<\gamma<3/2$, which covers the Coulomb and Yukawa interaction. The last is to extend the result for many components BEC, i.e., a system of $p$ components of bosons, each of which consists of $N_{1},N_{2},\ldots,N_{p}$ particles, respectively. The bosons are in three dimensions with interactions via a generalized interaction potential which includes the Coulomb interaction. We show that the optimal rate of convergence.